I’m now getting into the nitty gritty of Bindseil’s book on monetary policy implementation ("MPI" - my abbreviation), which is to say the part where we start to develop a mathematical view of MPI. The math in this post is limited to basic balance sheet accounting and algebra, but we’re working our way towards a more rigorous view. Again, as a reminder, this is a summary of Bindseil's book, infused and modified with my own interpretation and explanations.
Bindseil states the following basic principle of MPI, which I think provides succinct guidance for building a model of MPI:
Bindseil states the following basic principle of MPI, which I think provides succinct guidance for building a model of MPI:
“Influence through monetary policy instruments the demand and supply of reserves such that their price, namely, the overnight interbank interest rate, is close to the target level that has been defined according to the prevailing stance of monetary policy.” *What this quote tells us is that we’ll eventually need to model bank reserve supply and demand if we want to successfully model MPI. In this post, we’ll begin to do this by taking advantage of the central bank’s balance sheet, transforming it from an identity into a simple one-equation model with exogenous and endogenous variables. To be clear, we won’t have a complete model of MPI at the end of this post, but the equation that we will have will help us build intuition that will allow us to complete a more robust model in future posts.
Getting Familiar with the Central Bank Balance Sheet
Though maintaining his position that MPI normally means controlling interest rates as opposed to monetary quantities, Bindseil explains that the best starting-point for understanding MPI is to understand quantities on the central bank’s balance sheet. In order to analyze MPI, Bindseil suggests viewing the central bank’s balance sheet according to the following organization:
Autonomous liquidity factors – This is sort of a catch-all for all items on the balance sheet of the central bank that don’t reflect monetary policy operations or reserve holdings of banks with the central bank. Note that this classification may depend on the specific monetary policy regime adopted, as foreign currency or gold could be considered a core element of monetary policy operations under a fixed rate regime. However, under a floating rate regime, items outside of normal monetary policy operations would include issuing banknotes, foreign exchange reserves, investing funds to obtain income, float, conducting emergency liquidity assistance, paying out profits to government, etc. All of these transactions are not controlled by the monetary policy function of the central bank. Nonetheless, they need to be accounted for, since they affect the scarcity of bank reserves.
Monetary Policy Operations – These include balance sheet entries that reflect monetary policy operations, namely the use of open market operations ("OMOs") and standing facilities (lending and deposit facilities). OMOs are conducted at the initiative of the central bank in order to achieve its operational target of monetary policy. In contrast to OMOs, use of standing facilities are monetary policy operations conducted at the initiative of banks.
Central banks determine the interest rates they want to offer on their standing facilities. Typically, the interest rate on liquidity providing facilities are higher than the interest rate on liquidity absorbing facilities, introducing a form of discipline on the banking system. The two rates create a “corridor,” or a ceiling and a floor, within which the interbank rate fluctuates. If no liquidity absorbing facility is offered, then this is equivalent to the floor being 0%.
Commercial bank reserves – Not much explanation needed here. Bindseil states that this is “perhaps the most crucial single item of the balance sheet for monetary policy implementation since [it represents] the good of which the short-term market interest rate is the price.”
Viewing the Central Bank’s Balance Sheet as a List of Reserve Supply Factors
One place to start building our model of MPI is applying the basic balance sheet identity, which stipulates that the sum of assets must equal the sum of liabilities and equity. In the context of the above central bank balance sheet, this is represented in Equation 1.
Equation 1: Autonomous factor assets + OMO assets + Liquidity-injecting standing facility = Autonomous factor liabilities + OMO liabilities + Liquidity-absorbing standing facility + commercial bank reserves
Suppose we view reserves as a residual that balances the balance sheet identity. This is similar to using
“cash” on a firm’s balance sheet as the “plug” in a valuation model. We can rearrange the equation to reflect this viewpoint as such:
Equation 2: Commercial bank reserves (R) = net OMOs (M) + borrowing standing facility (B) – deposit standing facility (D) – net autonomous factors (A). Using variables instead of words: R = M + B – D – A
The word “net” consolidates corresponding assets and liabilities into an “asset – liability” sum, which makes the equation easier to read for our purposes. Monetary policy operations are being netted as an asset on the central bank balance sheet. Autonomous factors are being netted as a liability. The signs on the variables should thus make intuitive sense. If M is being taken as a net asset, then it should have a positive sign, since an increase in assets increases R, all else equal, for the balance sheet to balance. Conversely, D and A should have minus signs since as liabilities increase, R decreases, all else equal. To improve your understanding of this equation, it helps to make a simple spreadsheet modeled off of the balance sheet above. The cell for R should be programmed with the above equation, adapted to the variables’ corresponding cells. Then enter arbitrary values for the rest of the variables, and get comfortable with how and why the balance sheet balances.
Another way to describe Equation 2 is that it captures all the sources and drains of commercial bank reserves. In this sense, the central balance sheet provides a sort of supply function of reserves.
Okay, let’s now make practical use of this equation. Suppose our goal, as the central bank, is to stabilize the interbank rate for a given demand for reserves, and in doing so, to minimize recourse to standing facilities. By simple supply and demand analysis, we must therefore stabilize the supply of reserves R. Examining equation 2, this means we must offset changes in autonomous liquidity factors A with open market operations M. In other words, any change in A needs to be offset by a change in M in order to keep R constant. In the real world, central banks indeed devote considerable resources to forecasting changes in autonomous factors for this very purpose. Along these lines, we can consider autonomous factor changes A as an “exogenous” variable, or a quantity that will be given rather than determined by our model. More on this in the concluding section of this post.
Incorporating Reserve Demand Over the Reserve Maintenance Period
To complete a model of the market for reserves, we also need to capture the demand factors. Bindseil states that the main demand factors are captured in the concepts of required reserves and excess reserves. These concepts are summarized in Equation 3.
Equation 3: R = RR + XSR + RD (all variables should be interpreted as averages over the maintenance period)
RR stands for the quantity of required reserves, XSR for excess reserves, and RD for reserve deficiencies. The sum of these equals total reserves. RD occurs when banks do not fulfil their RR and must borrow at a penalty rate that is higher than what is provided by the borrowing standing facility (labeled as B in Equation 2). This is rare, and we’ll assume it away for simplicity. Technically, the quantities in Equation 3 should be specified as averages over the maintenance period, since in the real world, RR usually only need to be held on average over reserve maintenance period.
For those unfamiliar with the concept of a reserve maintenance period, this is a period in time during which banks must hold a certain quantity of reserves. This quantity is determined by the reserve requirement, which usually is a certain percentage of bank deposits held in a previous time period, called the reserve computation period. So for example, let's say banks have two weeks to go about their business without needing to maintain a certain amount of RR - the reserve computation period - and end up with 100 deposits. Let's say the reserve requirement is 10% of deposits and the maintenance period is the following two weeks. During this period of time, the bank thus needs to maintain 10 (10%*100) reserves on average at the end of each day. There are other ways to structure a reserve maintenance period, but this is how it currently works in the U.S. at a high level.
Let’s update Equation 2 in light of this added detail, by substituting in “RR + XSR” for R and reinterpreting the equation as describing average central bank balance sheet quantities over the reserve maintenance period:
Equation 4: RR + XSR = M + B – D – A (all variables should be interpreted as averages over the maintenance period)
Considering the concept of exogeneity again, Bindseil cites some of his own empirical research from euro area data in order to assert that demand for XSR can be treated as an exogenous demand factor in the context of day to day monetary policy operations. "Bindseil et al. (2004) show that there are no indications from euro area data that excess reserves would depend on liquidity conditions or on short-term interest rates. Therefore, in the weekly calibration of open market operations, they can effectively be treated as an exogenous demand factor, which needs to be forecast similarly to autonomous factors." Bindseil shows that the trend in XSR throughout a maintenance period increases monotonously, which is simply a consequence of banks that have already fulfilled reserve requirements experiencing end-of-day positive liquidity shocks (they cannot trade away the excess reserves in the interbank market, which has closed).
Reserve requirements RR, at least on a lagged basis, can much more easily be seen as exogenous, since banks must hit a given reserve quantity during the reserve maintenance period by hook or crook.
We’re now getting closer to a more developed model of the central bank’s balance sheet over the maintenance period. To continue developing it, let’s build some intuition by thinking about what might happen if reserve demand is not equal to a given reserve supply. Bindseil writes:
“Under the assumption of perfect interbank markets and large reserve requirements, recourse to standing facilities should take place in principle only at the end of the reserve maintenance period and should basically reflect the aggregate lack or surplus of accumulated reserves relative to the given demand for reserves… Also, this type of recourse to standing facilities should be definition always be one-sided, that is, a simultaneous aggregate recourse of the banking system to both a liquidity-providing and to a liquidity-absorbing standing facilities can never occur.”
Let’s unpack this. What Bindseil is saying is that any banks that wind up with reserves beyond that which they demand should trade them away in the interbank market to banks that are short of reserves. Lending the reserves is more profitable for the lending bank than holding the reserves, and borrowing the reserves in the interbank market is cheaper for the borrowing bank than going to the lending facility. Only at the end of the maintenance period, if banks have not accumulated sufficient reserves, should they seek recourse at the lending facility. This reflects an aggregate lack of accumulated reserves relative to the demand for reserves.
Or vice versa. If there are more reserves in the system than banks demand at the end of the maintenance period, then these will be placed into the liquidity-absorbing facility, if it exists.
Under these assumptions, you should only have one or the other – aggregate recourse to the lending or deposit facilities. Bindseil notes that this type of recourse “should be considered as a residual of the central bank balance sheet over the maintenance period.”
If we do not assume perfect markets, then we might see non-aggregate recourse to the central bank’s standing facilities. In other words, in the middle of the maintenance period, individual banks may place reserves into the liquidity-absorbing facility or borrow at the lending facility. Bindseil says:
“Such recourse has little to do with the aggregate liquidity situation, but mainly reflects transaction costs or failures in the payment systems and non-anticipated end-of-day payment follows which occur too late to allow a correction via the inter-bank market and which cannot be averaged out through reserve requirements.”
As such, the central bank can treat individual recourse as exogenous to the aggregate availability of reserves, similar to that of an autonomous liquidity factor.
Let’s now expand equation 4 above to distinguish between individual and aggregate recourse. Subscript “i” will stand for “individual,” and subscript “a” will stand for “aggregate.”
Equation 5: RR + XSR = M + Ba + Bi – Da – Di – A (all variables as averages over the maintenance period)
Completing a Simple, One-Equation Model of the Central Bank’s Balance over the Reserve Maintenance Period
To transform equation 5, which is more or less a balance sheet identity, into a basic modeling tool for monetary policy implementation during the reserve maintenance period, we need to specify which variables are to be exogenous and which are endogenous. To do this, we need to identify the residual of the balance sheet over the maintenance period. Defining “the residual of the balance sheet means identifying which balance sheets variables are to be considered exogenous and which are endogenous in the one-equation model which is a balance sheet.” In other words, the residual will be determined by the rest of the equation and will thus be endogenous. The rest of the variables will be exogenous.
If we’re modeling the balance sheet over the course of the reserve maintenance period, we’ve already recognized required reserves RR, autonomous liquidity factors A, individual recourse Bi and Di, and excess reserves XSR as exogenous. This leaves monetary policy operations M as well as aggregate recourse to the standing facilities Ba and Da as candidates for our model’s endogenous variables. Let’s then consider two cases in turn: one where M is given, and one where M is yet to be determined.
We can consider M as given, or exogenous, in a scenario where OMOs are conducted on a time-limited basis and “the last operation of the reserve maintenance period has already taken place.” If this is the case, then the residual of the balance sheet over the maintenance period is the net aggregate recourse to the standing facilities:
Equation 6: Ba – Da = RR + XSR - M - Bi + Di + A (all variables as averages over the maintenance period)
As discussed in the previous section, aggregate recourse at the end of the reserve maintenance period should be one-sided: one of Ba and Da will be positive and the other zero. Ba will be positive if the right side of the equation is positive; Da will be positive if the right side of the equation is negative. This makes sense if you think of Ba and Da as being the ultimate sources or drains of reserves to make the balance sheet balance.
For example, suppose RR = 10, and banks are determined to acquire 5 XSR. Over the course of the maintenance period, M net added only 10 reserves. Individual recourse borrowing added another 5 reserves. Individual deposit recourse soaked up 4 reserves. Autonomous factors net soaked up 4 reserves. The right side of Equation 6 would then read 10 + 5 – 10 – 5 + 4 + 4, which equals 8. One way to think about this is that during the maintenance period, 15 reserves are sourced from M and Bi. Perhaps 8 of these went to Di and A, and the leftover 7 go to fulfilling RR and XSR. But RR + XSR = 15, so 8 more reserves are still needed to make the equation balance. The only remaining place the reserve can come from is Ba, and Da will necessarily be 0.
If a deposit standing facility is not available (such as in the U.S. prior to 2008), then Equation 6 could be modified to
Equation 7: Ba – XSRa = RR + XSRi - M - Bi + Di + A (all variables as averages over the maintenance period)
where XSRa would include the quantity of reserves that would otherwise go to Da, and XSRi assumes the role of excess reserves (XSR) and individual recourse to the deposit facility (XSRi). In this case, either Ba or XSRa would be zero and the other positive.
The alternative case to consider is one in which OMOs are not (yet) given and still to be determined by the central bank. One major purpose of OMOs is to avoid aggregate recourse to standing facilities, since if that were to occur, market rates would be driven to the respective standing facility rate. With this in mind, we can set Ba and Da to zero and make M the endogenous variable. This means M is to be determined by the rest of the balance sheet.
Equation 8: M = RR + E[XSR + A + Di – Bi | It] (all variables as averages over the maintenance period)
E[. | It] is “an operation indicating expectations by the central bank based on available information at time t of the allotment decision.”
There will however be some point at which the last OMO is given, making M exogenous. In this case, Ba - Da becomes endogenous again. We can however augment Equation 6 (as well as Equation 7, though I do not do that here) with the expectations operator to incorporate the “previous endogeneity” of M:
Equation 9: Ba – Da = XSR + A - Bi + Di - E[XSR + A + Di – Bi | It] (all variables as averages over the maintenance period)
To arrive at Equation 9, simply substitute the M in Equation 6 with RR + E[XSR + A + Di – Bi | It] from Equation 8. It happens that RR cancels out, which is obvious algebraically, but also makes sense intuitively, since the central bank can be sure about the value of RR when it sets M. The other variables remain in Equation 9 however, since these values are uncertain.
Equation 9 is our ultimate destination in terms of understanding the central bank balance sheet over the reserve maintenance period, at least at this level of complexity. Bindseil describes Equation 9 as such:
“In words: from an ex post perspective, the balance sheet residual is the net aggregate recourse to standing facilities; it is determined by the difference between the actual exogenous liquidity factors and the central bank’s forecast of these factors at the moment of the (last) open market operation of the reserve maintenance period.”
In following posts, we'll model how the central bank controls interest rates, using the intuition developed in this post, particularly from Equation 6.
*Note how the concepts of monetary policy implementation and monetary policy strategy are being invoked in this principle. Their use is subtle, but the distinction is critical. Strategy is defining the interest rate we want to achieve, whereas implementation is the means by which we achieve that interest rate. When we model MPI, we are taking strategy, and thus the target interest rate, as given and purely focusing on hitting that interest rate.